Positive semi-definite matrices, exponential convexity for multiplicative majorization and related means of Cauchy's type

In this paper, we obtain new results concerning the generalizations of additive and multiplicative majorizations by means of exponential convexity. We prove positive semi-definiteness of matrices generated by differences deduced from majorization type results which implies exponential convexity and \(\log\)-convexity of these differences and also obtain Lyapunov's and Dresher's inequalities for these differences. We give some applications of additive and multiplicative majorizations. In addition, we introduce new means of Cauchy's type and establish their monotonicity